Question

1. mindy’s high school class has a total of 150 students, and they are planning a trip. the school has small buses and vans to transport all students. each van can seat 6 students, and each bus can seat 14 students. there are only 15 drivers, so the total number of vehicles that they can take must be 15. how many of each vehicle will mindy’s class need to transport everyone? (a) write a system of equations to describe the situation. (b) find the solution to the system, and answer the question.

Explanation:

Step1: Define variables

Let $v$ = number of vans, $b$ = number of buses.

Step2: Set up vehicle count equation

Total vehicles = 15:
$v + b = 15$

Step3: Set up student capacity equation

Total students = 150, vans hold 8, buses hold 14:
$8v + 14b = 150$

Step4: Solve first equation for $v$

$v = 15 - b$

Step5: Substitute into second equation

$8(15 - b) + 14b = 150$
$120 - 8b + 14b = 150$
$120 + 6b = 150$

Step6: Solve for $b$

$6b = 150 - 120$
$6b = 30$
$b = \frac{30}{6} = 5$

Step7: Solve for $v$

$v = 15 - 5 = 10$

Answer:

(a) System of equations:

$v + b = 15$
$8v + 14b = 150$

(b) Solution:

10 vans and 5 buses are needed.